Combinatorial Game Theory
who wins, and by how much
Generator

One move, in each game

One move, in each game
One move, in each game. A move joins two unused ends with a curve that crosses nothing, and puts a new spot on it. In Brussels Sprouts the ends are the drawn arms of a cross and the new spot is a crossbar, so two ends go and two arrive. In Sprouts the ends are unused lives of a dot and the new spot has three of them, two of which the curve has just spent — so two go and one arrives.

A move joins two unused ends with a curve that crosses nothing, and puts a new spot on it. In Brussels Sprouts the ends are the drawn arms of a cross and the new spot is a crossbar, so two ends go and two arrive. In Sprouts the ends are unused lives of a dot and the new spot has three of them, two of which the curve has just spent — so two go and one arrives.

1 essay calls sprouts-move. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

One of these is a game. Every length that came out of 40 random games from each starting position. Brussels Sprouts always ends after exactly five crosses less two moves, so whoever is to move at that point was decided before the first curve was drawn. Sprouts ends at different lengths depending on how it is played, which is what makes it worth playing. Particular games

Sprouts, and the game that is not one

Two games played with dots and curves, invented in the same room, all but indistinguishable on paper. One is unsolved past forty spots. The other has no decisions in it at all — the winner is fixed before the first curve is drawn.

The whole library · The position index · The figures that play back